Retardation Plates

This is a continuation from the previous tutorial - non-normal-incidence reflection and transmission polarizers.

1. Introduction

The theory of retardation plates and especially quarter-wave retarders is given in later tutorial series. The basic relation for retardation plates is

\[\tag{9}N\lambda=d(n_e-n_o)\]

where \(n_o\) is the refractive index of the ordinary ray, \(n_e\) is the refractive index of the extraordinary ray, \(d\) is the physical thickness of the plate, and \(\lambda\) is the wavelength.

Retardation plates are generally made of mica, stretched polyvinyl alcohol, and quartz, although other stretched plastics such as cellophane, Mylar, cellulose acetate, cellulose nitrate, sapphire, magnesium fluoride, and other materials can also be used.

Polyvinyl alcohol in sheet form transmits well into the ultraviolet beyond the cutoff for natural mica and is thus particularly useful for ultraviolet retardation plates.

As suggested by Jacobs et al., permanent birefringence can be thermomechanically induced in the borosilicate optical glass ARG-2, making it an attractive alternate to natural crystalline quartz and mica for large aperture wave plates for laser systems.

Refractive indexes and birefringences of some materials are listed in Tables 7 and 8.

The birefringences reported for mica and apophyllite should be considered as approximate, since they are measurements made on single samples. There is good reason to believe that the birefringence of apophyllite may be different for other samples.

Although calcite would seem at first to be a good material for retardation plates, its birefringence is so high that an extremely thin piece, less than 1 μm, would be required for a single \(\lambda/4\) retardation plate. If a ‘‘first-order’’ or multiple-order plate were constructed, or if calcite were used as one component of an achromatic retardation plate, the tolerance on the thickness would be very stringent.

Table 8.  Birefringence \(n_e-n_o\) of Various Optical Materials

Retardation plates are generally made of a single piece of material, although when the thickness required for a plate is too small, two thicker pieces may be used with the fast axis of one aligned parallel to the slow axis of the other to cancel out all but the desired retardation.

Plates which are a little too thin or a little too thick may be rotated about an axis parallel or perpendicular to the optic axis to change the retardation to the desired amount, as suggested by Gieszelmann et al., and Daniels.

There are also some novel circular polarizers and polarization rotators for use in the far ultraviolet, far infrared, and visible region.

Achromatic retardation plates which have the same retardation over a range of wavelengths can be made from two or more different materials or from two or more plates of the same material whose axes are oriented at appropriate angles with respect to each other.

These latter devices are known as composite plates, and although they can change plane-polarized light into circularly polarized light, they do not have all the other properties of true retardation plates. By far the most achromatic \(\lambda/4\) retarders are devices, such as the Fresnel rhomb, which obtain their retardation from internal reflections at angles greater than the critical angle.

Methods for making and testing quarter-wave plates including ways of splitting mica, how to distinguish between fast and slow axes, methods for measuring retardations close to \(\lambda/4\), and the tolerance on plate thickness will be described in detail in later tutorials. An additional paper by Nakadate shows how Young’s fringes can be used for a highly precise measurement of phase retardation.

Waveplates are all sensitive to some degree to temperature changes, variations in the angle of incidence, coherence effects in the light beam, and wavelength variations. Multiple-order plates are much more sensitive than ‘‘first-order’’ or single-order plates. Hale and Day discuss these effects for various types of waveplates and suggest designs that are less sensitive to various parameters.

Most retardation plates are designed to be used in transmission, generally at normal incidence. However, there are also reflection devices that act as quarter-wave and half-wave retarders and polarization rotators.

In the vacuum ultraviolet, Westerveld et al. produced circularly polarized light by using Au-coated reflection optics. Saito et al. used an evaporated Al mirror as a retardation plate at 1216 Å, Lyman \(\alpha\) radiation, following earlier work by McIlrath.

Greninger showed that a three-mirror device could be used in place of a half-wave plate to rotate the plane of polarization of a plane-polarized beam and preserve the collinearity of input and output beams. Johnston used a different three-mirror arrangement for the same application in the far-infrared.

Thonn and Azzam designed three-reflection half-wave and quarter-wave retarders from single-layer dielectric coatings on metallic film substrates. They showed calculations for ZnS-Ag film-substrate retarders used at 10.6 μm.

Previously Zaghloul, Azzam, and Bashara had proposed using a SiO₂ film on Si as an angle-of-incidence tunable reflection retarder for the 2537-Å mercury line in the ultraviolet spectral region. Kawabata and Suzuki showed that a film of MgF₂ on Ag was superior to Zaghloul et al.’s design at 6328 Å. They also performed calculations using Al, Cu, and Au as the metals and concluded that Ag worked best.

Mica Retardation Plates

Mica quarter-wave plates can be made by splitting thick sheets of mica down to the appropriate thickness. Since the difference between the velocities of the ordinary and extraordinary rays is very small, the mica sheets need not be split too thin; typical thicknesses lie in the range 0.032 to 0.036 mm for yellow light.

The fast and slow axes of a mica quarter-wave plate can be distinguished using Tutton’s test and the retardation can be measured using one of several rather simple tests.

If the mica sheets are used without glass cover plates, multiply reflected beams in the mica can cause the retardation to oscillate around the value calculated from the simple theory. Fortunately this effect can be eliminated in one of several ways.

Mica does have one serious drawback. There are zones in the cleaved mica sheets which lie at angles to each other and which do not extinguish at the same angle. Thus, extinction cannot be obtained over the whole sheet simultaneously.

In very critical applications such as ellipsometry, much better extinction can be obtained using quarter-wave plates made of crystalline quartz, which do not exhibit this effect.

2. Crystalline-Quartz Retardation Plates

Crystalline quartz is also frequently used for retardation plates, particularly those of the highest quality. It escapes the problem of zones with different orientations like those found in mica.

The thickness of quartz required for a single quarter-wave retardation at the 6328-Å helium-neon laser line is about 0.017 mm, much too thin for convenient polishing. If the plate is to be used in the infrared, single-order quarter-wave plates are feasible.

Two types of quartz retardation plates are generally employed in the visible and ultraviolet regions: so-called ‘‘first-order’’ plates made of two pieces of material, which are the best for critical applications, and multiple-order plates made of one thick piece of crystalline quartz.

The multiple-order plates are generally not used for work of the highest accuracy since they are extremely sensitive to small temperature changes and to angle of incidence. Also, they have \(\lambda/4\) retardation only at certain wavelengths; at other wavelengths the retardation may not even be close to \(\lambda/4\).

‘‘First-Order’’ Plates

A so-called ‘‘first-order’’ plate is made by cementing together two nearly equal thicknesses of quartz such that the fast axis of one is aligned parallel to the slow axis of the other (both axes lie in planes parallel to the polished faces).

The plate is then polished until the difference in thickness between the two pieces equals the thickness of a single \(\lambda/4\) plate. The retardation of this plate can be calculated from Eq. (9) by setting \(d\) equal to the difference in thickness between the two pieces.

The ‘‘first-order’’ plate acts strictly like a single-order quarter-wave plate with respect to the variation of retardation with wavelength, temperature coefficient of retardation, and angle of incidence.

The change in phase retardation with temperature at 6328 Å is \(0.0091^\circ/^\circ\text{C}\), less than one-hundredth that of the 1.973-mm multiple-order plate. The change in retardation with angle of incidence at this wavelength is also small: \((\Delta{N})_{10^\circ}=0.0016\), as compared with 0.18 for the thick plate.

A ‘‘first-order’’ quartz \(\lambda/4\) plate has several advantages over a mica \(\lambda/4\) plate.

1. Crystalline quartz has a uniform structure, so that extinction can be obtained over the entire area of the plate at a given angular setting.
2. Since the total plate thickness is generally large, of the order of 1 mm or so, the coherence of the multiple, internally reflected beams is lost and there are no oscillations in the transmitted light or in the phase retardation.
3. Crystalline quartz is not pleochroic, except in the infrared, so that the intensity transmitted along the two axes is the same.
4. Crystalline quartz transmits farther into the ultraviolet than mica, so that ‘‘first-order’’ plates can be used from about 0.185 to 2.0 μm (see Table 8).

Single-Order Plates in the Infrared

Although a crystalline-quartz retardation plate which is \(\lambda/4\) in the visible is too thin to make from a single piece of material, the thickness required for such a plate is larger in the infrared.

Jacobs and coworkers describe such a \(\lambda/4\) plate for use at the 3.39-μm helium-neon laser line. They measured the birefringence of quartz at this wavelength and found it to be \(0.0065\pm 0.0001\), so that the thickness required for the plate was 0.1304 mm. The actual plate was slightly thinner (0.1278 mm), so that it was tipped at an angle of \(10^\circ\) (rotating it about an axis parallel to the optic axis) to give it exactly \(\lambda/4\) retardation.

Maillard has also measured the birefringence of quartz at 3.39 and 3.51 μm and obtained values of 0.00659 and 0.00642, respectively (both \(\pm0.00002\)), in agreement with Jacobs’ value.

A problem encountered when using crystalline quartz in the infrared is that, in general, the ordinary and extraordinary rays have different absorption coefficients; thus it may be impossible to construct a perfect wave plate regardless of the relative retardation between the rays.

For an absorbing wave plate to have a retardation of exactly \(\lambda/4\), the requirement

\[\tag{10}\left(\frac{n_o+1}{n_e+1}\right)^2\exp\left[-\frac{(\alpha_e-\alpha_o)\lambda}{8(n_e-n_o)}\right]=1\]

must be met; \(\alpha_e\) and \(\alpha_o\) are the absorption coefficients for the extraordinary and ordinary rays, respectively.

At wavelengths shorter than 3.39 μm, the birefringence is small enough for it to be possible to approximate the condition in Eq. (10) closely whenever \(\alpha_e\approx\alpha_o\).

Gonatas et al. concluded that, in the far infrared and submillimeter wavelength region, the effect of different absorption coefficients in the crystalline quartz was small and could be corrected for.

Another problem which occurs for crystalline quartz and also for sapphire in the infrared is that the Fresnel reflection coefficients are slightly different for the ordinary and extraordinary rays since the refractive indexes and absorption coefficients are in general different.

One possible solution is to deposit isotropic thin films on the crystal surfaces. The refractive index of these films is chosen to balance the anisotropic absorption effect by making the Fresnel reflection coefficients appropriately anisotropic. On the other hand, if anisotropic Fresnel reflection proves to be undesirable, it can be greatly diminished by using an antireflection coating.

If a single-order, crystalline-quartz plate is to be used for a continuous range of wavelengths, both the phase retardation and the transmittance of the ordinary and extraordinary rays will oscillate as a function of wavelength because of multiple coherent reflections in the quartz.

Thus, if a wave plate is to be used over a range of wavelengths, it would be well to antireflect the surfaces to eliminate the phase oscillations.

Multiple-Order Plates

Thick plates made from crystalline quartz are sometimes used to produce circularly polarized light at a single wavelength or a discrete series of wavelengths. The plate thickness is generally of the order of one or more millimeters so that the retardation is an integral number of wavelengths plus \(\lambda/4\), hence the name multiple-order wave plate.

This plate acts like a single \(\lambda/4\) plate providing it is used only at certain specific wavelengths; at other wavelengths it may not even approximate the desired retardation.

For example, a 1.973-mm-thick quartz plate was purchased which had an order of interference \(N=28.25\) at 6328 Å. From Eq. (9) and Table 8, this plate would have \(N=30.52\) at 5890 Å, and would thus be an almost perfect half-wave plate at this latter wavelength.

If a multiple-order plate is used to produce circularly polarized light at unspecified discrete wavelengths, e. g., to measure circular or linear dichroism, it can be placed following a polarizer and oriented at \(45^\circ\) to the plane of vibration of the polarized beam. When the wavelengths are such that \(N\) calculated from Eq. (9) equals 1/4, 3/4, or in general \((2M-1)/4\) (where \(M\) is a positive integer), the emerging beam will be alternately right and left circularly polarized.

The frequency interval \(\Delta\nu\) between wavelengths at which circular polarization occurs is

\[\tag{11}\Delta\nu=\frac{1}{2d(n_e-n_o)}\]

where \(\nu=1/\lambda\).

If the birefringence is independent of wavelength, the retardation plate will thus produce circularly polarized light at equal intervals on a frequency scale and can conveniently be used to measure circular dichroism.

In order to approximately calibrate a multiple-order retardation plate at a series of wavelengths, it can be inserted between crossed polarizers and oriented at \(45^\circ\) to the polarizer axis. Transmission maxima will occur when the plate retardation is \(\lambda/2\) or an odd multiple thereof; minima will occur when the retardation is a full wave or multiple thereof.

If the axes of the two polarizers are parallel, maxima in the transmitted beam will occur when the plate retardation is a multiple of a full wavelength. The birefringence of the retardation plate can be determined by measuring the wavelengths at which maxima or minima occur if the plate thickness is known. Otherwise \(d\) can be measured with a micrometer, and an approximate value of \(n_e-n_o\) can be obtained.

Palik made and tested a 2.070-mm-thick CdS plate for the 2- to 15-μm infrared region and also made thick retardation plates of SnSe, sapphire, and crystalline quartz to be used in various parts of the infrared. Holzwarth used a cultured-quartz retardation plate 0.8 mm thick to measure circular dichroism in the 1850- to 2500-Å region of the ultraviolet; Jaffe et al. measured linear dichroism in the ultraviolet using a thick quartz plate and linear polarizer.

Sensitivity to Temperature Changes

Small temperature changes can have a large effect on the retardation of a multiple-order plate. For the 1.973-mm-thick quartz plate (\(N=28.25\) at 6328 Å), the phase retardation will decrease \(1.03^\circ\) for each Celsius degree increase in temperature.

If the temperature of the wave plate is not controlled extremely accurately, the large temperature coefficient of retardation can introduce sizable errors in precise ellipsometric measurements in which polarizer and analyzer settings can be made to \(\pm0.01^\circ\).

Sensitivity to Angle of Incidence

The effect of angle of incidence (and hence field angle) on the retardation was calculated. It was shown that the change in phase retardation with angle of incidence, \(2\pi(\Delta{N})_{\theta}\), is proportional to the total thickness of the plate (which is incorporated into \(N\)) and the square of the angle of incidence when the rotation is about an axis parallel to the optic axis.

If the 1.973-mm-thick plate mentioned previously is rotated parallel to the optic axis through an angle of \(10^\circ\) at a wavelength of 6328 Å, the total retardation changes from 28.25 to 28.43 , so that the \(\lambda/4\) plate is now nearly a \(\lambda/4\) plate.

If the plate had been rotated about an axis perpendicular to the direction of the optic axis, in the limit when the angle of incidence is \(90^\circ\), the beam would have been traveling along the optic axis; in this case the ordinary and extraordinary rays would be traveling with the same velocities, and there would have been no retardation of one relative to the other.

For any intermediate angle of incidence the retardation would have been less than the value at normal incidence. The relation for the retardation as a function of angle of incidence is not simple, but the retardation will be approximately as angle-sensitive as it was in the other case.

An advantage of rotation about either axis is that, with care, one can adjust the retardation of an inexact wave plate to a desired value. Rotation about an axis parallel to the optic axis will increase the retardation, while rotation about an axis perpendicular to the optic axis will decrease the retardation.

3. Achromatic Retardation Plates

Achromatic retardation plates are those for which the phase retardation is independent of wavelength. The name arose because when a plate of this type is placed between polarizers, it does not appear colored and hence is achromatic.

In many applications, a truly achromatic retardation plate is not required. Since the wavelength of light changes by less than a factor of 2 across the visible region, a quarter- or half-wave mica plate often introduces only tolerable errors even in white light. The errors that do occur cancel out in many kinds of experiments.

Achromatic retardation plates can be made in various ways. The most achromatic are based on the principle of the Fresnel rhomb, in which the phase retardation occurs when light undergoes two or more total internal reflections.

A material with the appropriate variation of birefringence with wavelength can also be used. Such materials are uncommon, but plates of two or more different birefringent materials can be combined to produce a reasonably achromatic combination.

Composite plates, consisting of two or more plates of the same material whose axes are oriented at the appropriate angles, can be used as achromatic circular polarizers or achromatic polarization rotators, although they do not have all the properties of true \(\lambda/4\) or \(\lambda/2\) plates.

The simplest type of achromatic retardation plate could be made from a single material if its birefringence satisfied the requirement that \((n_e-n_o)/\lambda\) be independent of wavelength, i.e., that \(n_e-n_o\) be directly proportional to \(\lambda\).

This result follows from Eq. (9) since \(d(n_e-n_o)/\lambda\) must be independent of \(\lambda\) to make \(N\) independent of wavelength. (The plate thickness \(d\) is constant.)

The birefringences of various materials are listed in Table 8 and plotted in Figs. 18 and 19. Only one material, the mineral apophyllite, has a birefringence which increases in the correct manner with increasing wavelength.

A curve of the phase retardation vs. wavelength for a quarter-wave apophyllite plate is shown as curve \(D\) in Fig. 20. Also included are curves for other so-called achromatic \(\lambda/4\) plates as well as for simple \(\lambda/4\) plates of quartz and mica.

The phase retardation of apophyllite is not as constant with \(\lambda\) as that of the rhomb-type retarders, but it is considerably more constant than that of the other ‘‘achromatic’’ \(\lambda/4\) plates.

Since the birefringence of apophyllite is small, a \(\lambda/4\) plate needs a thickness of about 56.8 μm, which is enough for it to be made as a single piece rather than as a ‘‘first-order’’ plate.

Unfortunately optical-grade apophyllite is rare, the sample for which data are reported here having come from Sweden. There is some indication that the optical properties of other apophyllite samples may be different.

Isotropic, positive, and negative-birefringent specimens have been reported by Deer et al. According to them, the optical properties of apophyllite are often anomalous, some specimens being isotropic, uniaxial negative, or even biaxial with crossed dispersion of optic axial planes. Whether many samples have the favorable birefringence of the Swedish sample is uncertain.

Certain types of plastic film stretched during the manufacturing process have birefringences which are nearly proportional to wavelength and can serve as achromatic retardation plates if they have the proper thickness.

Curve C in Fig. 20 is the retardation of a stretched cellulose nitrate film. A combination of stretched cellulose acetate and cellulose nitrate sheets with their axes parallel will also make a reasonably achromatic \(\lambda/4\) plate over the visible region.

The advantages of using stretched plastic films for retardation plates are that they are cheap, readily available, have a retardation which is uniform over large areas, and can be used in strongly convergent light.

However, each sheet must be individually selected since the birefringence is a strong function of the treatment during the manufacturing process and the sheets come in various thicknesses, with the result that their retardations are not necessarily \(\lambda/4\) or \(\lambda/2\).

Also, Ennos found that while the magnitude of the retardation was uniform over large areas of the sheets, the direction of the effective crystal axis varied from point to point by as much as \(1.5^\circ\) on the samples he was testing. Thus, film retarders appear to be excellent for many applications but are probably not suitable for measurements of the highest precision.

A reasonably achromatic retardation plate can be constructed from pairs of readily available birefringent materials such as crystalline quartz, sapphire, magnesium fluoride, calcite, or others whose birefringences are listed in Table 8.

Assume that the plate is to be made of materials \(a\) and \(b\) having thicknesses \(d_a\) and \(d_b\), respectively (to be calculated), and that it is to be achromatized at wavelengths \(\lambda_2\) and \(\lambda_2\). From Eq. (9) we can obtain the relations

\[\tag{12}\begin{align}N\lambda_1&=d_a\Delta{n}_{1a}+d_b\Delta{n}_{1b}\\N\lambda_2&=d_a\Delta{n}_{2a}+d_b\Delta{n}_{2b}\end{align}\]

where \(N=\frac{1}{4}\) for a \(\lambda/4\)plate, \(\frac{1}{2}\) for a \(\lambda/2\) plate, etc., and the \(\Delta{n}\)'s are values of \(n_e-n_o\) for the particular materials at the wavelengths specified; \(\Delta{n}\) will be positive for a positive uniaxial crystal and negative for a negative uniaxial crystal. (A positive uniaxial material can be used with its fast axis crossed with that of another positive uniaxial material; in this case the first material will have a negative \(\Delta{n}\).)

Equations (12) can be solved for \(d_a\) and \(d_b\):

\[\tag{13}d_a=\frac{N(\lambda_1\Delta{n}_{2b}-\lambda_2\Delta{n}_{1b})}{\Delta{n}_{1a}\Delta{n}_{2b}-\Delta{n}_{1b}\Delta{n}_{2a}}\qquad{d_b}=\frac{N(\lambda_2\Delta{n}_{1a}-\lambda_1\Delta{n}_{2a})}{\Delta{n}_{1a}\Delta{n}_{2b}-\Delta{n}_{1b}\Delta{n}_{2a}}\]

As an example of a compound plate, let us design a \(\lambda/4\) plate of crystalline quartz and calcite and achromatize it at wavelengths \(\lambda_1=0.508\) μm and \(\lambda_2=0.656\) μm.

Quartz has a positive birefringence and calcite a negative birefringence (Table 8) so that \(Delta{n}_{1a}\) and \(Delta{n}_{2a}\) (for quartz) are positive and \(\Delta{n}_{1b}\) and \(\Delta{n}_{2b}\) (for calcite) are negative.

Equations (13) are satisfied for \(d_\text{qtz}=426.2\) μm and \(d_\text{calc}=21.69\) μm; thus the phase retardation is exactly \(90^\circ\) at these two wavelengths.

An equation of the form of those in Eqs. (12) is now used to calculate \(N\) for all wavelengths in the visible region using birefringence values listed in Table 8, and the results are plotted as curve E in Fig. 20.

Although the achromatization for this quartz-calcite combination is not as good as can be obtained with a rhomb-type device or apophyllite, the phase retardation is within \(\pm5^\circ\) of \(90^\circ\) in the wavelength region 4900 to 7000 Å and is thus much more constant than the retardation of a single mica or quartz \(\lambda/4\) plate.

Better two-plate combinations have been calculated by Beckers, the best being MgF₂-ADP and MgF₂-KDP, which have maximum deviations of \(\pm0.5\) and \(\pm0.4\) percent, respectively, compared with \(\pm7.2\) percent for a quartz-calcite combination over the same 4000- to 7000-Å wavelength region.

The thicknesses of the materials which are required to produce \(\lambda/4\) retardation are \(d_{MgF_2}=113.79\) μm, \(d_{ADP}=26.38\) μm, and \(d_{MgF_2}=94.47\) μm, \(d_{KDP}=23.49\) μm. Since the ADP and KDP must be so thin, these components could be made in two pieces as ‘‘first-order’’ plates.

If better achromatization is desired and one does not wish to use a rhomb-type \(\lambda/4\) device, three materials can be used which satisfy the relations

\[\tag{14}\begin{align}N\lambda_1&=d_a\Delta{n}_{1a}+d_b\Delta{n}_{1b}+d_c\Delta{n}_{1c}\\N\lambda_2&=d_a\Delta{n}_{2a}+d_b\Delta{n}_{2b}+d_c\Delta{n}_{2c}\\N\lambda_3&=d_a\Delta{n}_{3a}+d_b\Delta{n}_{3b}+d_c\Delta{n}_{3c}\end{align}\]

where the \(\Delta{n}\)’s are birefringences of the various materials at wavelengths \(\lambda_1\), \(\lambda_2\), and \(\lambda_3\).

Instead of using only three wavelengths, Beckers suggested that the thicknesses can be optimized such that the maximum deviations from achromatization are minimized over the entire wavelength interval desired.

In this way, he obtained a three-component combination of quartz, calcite, and MgF₂ which has a retardation of a full wavelength and a maximum deviation of only \(\pm0.2\) percent over the 4000- to 7000-Å wavelength region.

The maximum deviation of slightly different thicknesses of these same three materials rises to \(\pm2.6\) percent if the wavelength interval is extended to 3000 to 11,000 Å.

Chandrasekharan and Damany have designed a three-component \(\lambda/4\) plate from quartz, MgF₂, and sapphire for use in the vacuum ultraviolet.

Title has designed achromatic combinations of three-element, four-element, nine-element, and ten-element waveplates using Jones matrix techniques. The nine-element combination is achromatic to within \(1^\circ\) from 3500 to 10,000 Å. He constructed and tested several waveplate combinations, and they performed as designed.

4. Rhombs as Achromatic \(\lambda/4\) Retarders

The simplest stable, highly achromatic \(\lambda/4\) retarder with a reasonable acceptance angle and convenient size appears to be a rhomb-type retarder.

Several types are available; the choice of which one to use for a specific application depends on

1. The geometry of the optical system (can a deviated or displaced beam be tolerated?)
2. Wavelength range
3. Degree of collimation of the beam
4. Beam diameter (determining the aperture of the retarder)
5. Space available
6. Accuracy required.

Table 9 summarizes the properties of the various achromatic rhombs.

Anderson has compared the retardation of a CdS \(\lambda/4\) plate and a Fresnel rhomb in the 10-μm CO2 laser emission region. Wizinowich used a Fresnel rhomb along with some additional optics to change an unpolarized light beam from a faint star object into linearly polarized light to improve the throughput of a grating spectrograph and make it independent of the input polarization.

 

5. Composite Retardation Plates

A composite retardation plate is made up of two or more elements of the same material combined so that their optic axes are at appropriate angles to each other.

Some of the composite plates have nearly all the properties of a true retardation plate, whereas others do not.

Composite plates were described which produced circularly polarized light at a given wavelength, those which acted as achromatic circular polarizers, and those which acted as achromatic polarization rotators or pseudo \(\lambda/2\) plates.

The effect of combining several birefringent plates with their axes at arbitrary angles to each other can be easily understood using the Poincaré sphere.

6. Variable Retardation Plates and Compensators

Variable retardation plates can be used to modulate or vary the phase of a beam of plane-polarized light, to measure birefringence in mineral specimens, flow birefringence, or stress in transparent materials, or to analyze a beam of elliptically polarized light such as might be produced by transmission through a birefringent material or by reflection from a metal or film-covered surface.

The term compensator is frequently applied to a variable retardation plate since it can be used to compensate for the phase retardation produced by a specimen.

Common types of variable compensators include

• Babinet and Soleil compensators, in which the total thickness of birefringent material in the light path is changed.
• Sénarmont compensator, which consists of a fixed quarter-wave plate and rotatable analyzer to compensate for varying amounts of ellipticity in a light beam.
• Tilting-plate compensators, with which the total thickness of birefringent material in the light beam is changed by changing the angle of incidence.

Electro-optic and piezo-optic modulators can also be used as variable retardation plates since their birefringence can be changed by varying the electric field or pressure.

However, they are generally used for modulating the amplitude, phase, frequency, or direction of a light beam, in particular a laser beam, at frequencies too high for mechanical shutters or moving mirrors to follow.

Babinet Compensator

There are many devices which compensate for differences in phase retardation by having a variable thickness of a birefringent material (such as crystalline quartz) in the light beam. One such device can compensate for a residual wedge angle between the entrance and exit faces of birefringent optical components such as optical modulators and waveplates.

The most common variable retardation plates are the Babinet compensator and the Soleil compensator. The Babinet compensator was proposed by Babinet in 1837 and later modified by Jamin.

The Babinet compensator, shown schematically in Fig. 21, consists of two crystalline-quartz wedges, each with its optic axis in the plane of the face but with the two optic axes exactly \(90^\circ\) apart.

One wedge is stationary, and the other is movable by means of a micrometer screw in the direction indicated by the arrow, so that the total amount of quartz through which the light passes can be varied uniformly.

In the first wedge, the extraordinary ray vibrates in a horizontal plane and is retarded relative to the ordinary ray (crystalline quartz has a positive birefringence; see Table 8).

When the rays enter the second wedge, the ray vibrating in the horizontal plane becomes the ordinary ray and is advanced relative to the ray vibrating in the vertical plane. Thus , the total retardation is proportional to the difference in thickness between the two wedges:

\[\tag{15}N\lambda=(d_1-d_2)(n_e-n_o)\]

where \(N\) is retardation in integral and fractional parts of a wavelength, \(d_1\), \(d_2\) are the thickness of the first and second wedges where light passes through, and \(n_o\), \(n_e\) are the ordinary and extraordinary refractive indexes for crystalline quartz.

If light polarized at an angle of \(45^\circ\) to one of the axes of the compensator passes through it, the field will appear as shown in Fig. 21; the wedges have been set so there is zero retardation at the center of the field. (If the angle a of the incident plane-polarized beam were different from \(45^\circ\), the beam retarded or advanced by \(180^\circ\) in phase angle would make an angle of \(2\alpha\) instead of \(90^\circ\) with the original beam.)

When an analyzer whose axis is crossed with that of the polarizer is used to observe the beam passing through the compensator, a series of light and dark bands is observed in monochromatic light.

In white light only one band, that for which the retardation is zero, remains black. All the other bands are colored. These are the bands for which the retardation is multiples of \(2\pi\) (or, expressed in terms of path differences, integral numbers of wavelengths).

On one side of the central black band one ray is advanced in phase relative to the other ray; on the other side it is retarded. If one wedge is moved, the whole fringe system translates across the field of view. The reference line is scribed on the stationary wedge so that it remains in the center of the field.

Soleil Compensator

The Soleil compensator, sometimes called a Babinet-Soleil compensator, is shown in Fig. 22.

It is similar to the Babinet compensator in the way it is used, but instead of having a field crossed with alternating light and dark bands in monochromatic light, the field has a uniform tint if the compensator is constructed correctly. This is because the ratio of the thicknesses of the two quartz blocks (one composed of a fixed and a movable wedge) is the same over the entire field.

The Soleil compensator will produce light of varying ellipticity depending on the position of the movable wedge. Calibration of the Soleil compensator is similar to that of the Babinet compensator.

The zero-retardation position is found in the same manner except that now the entire field is dark. The compensator is used in the same way as a Babinet compensator with the uniformly dark field (in white light) of the Soleil corresponding to the black zero-retardation band in the Babinet .

The major advantage of the Soleil compensator is that a photoelectric detector can be used to make the settings. The compensator is offset a small amount on each side of the null position so that equal-intensity readings are obtained. The average of the two drum positions gives the null position.

Photoelectric setting can be much more precise than visual setting, but this will not necessarily imply increased accuracy unless the compensator is properly constructed.

Since Soleil compensators are composed of three pieces of crystalline quartz, all of which must be very accurately made, they are subject to more optical and mechanical defects than Babinet compensators.

Ives and Briggs found random departures of about \(\pm1.5^\circ\) from their straight-line calibration curve of micrometer reading for extinction vs. wedge position. This variation was considerably larger than the setting error with a half-shade plate and was attributed to variations in thickness of the order of \(\pm\lambda/4\) along the quartz wedges.

Soleil compensators have been used for measurements of retardation in the infrared. They have been made of crystalline quartz, cadmium sulfide, and magnesium fluoride. A by-product of this work was the measurement of the birefringence of these materials in the infrared.

Two other uniform-field compensators have been proposed. Jerrard has taken the Babinet wedges and reversed one of them so the light passes through the thicker portions of each wedge. This reversed Babinet compensator is less subject to mechanical imperfections than the Soleil compensator but does produce a small deviation of the main beam. Hariharan and Sen suggest double-passing a Babinet compensator (with a reflection between the two passes) to obtain a uniform field. 

The next tutorial discusses in detail about rare earth-doped fibers.